Complexity of term representations of finitary functions

Aichinger, Erhard; Mudrinski, Nebojša; Opršal, Jakub

doi:10.1142/s0218196718500480

Cited by 12 publications

(6 citation statements)

References 18 publications

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“…Also, for some algebras it makes sense to restrict the input to terms of a certain canonical form (for instance [18], [9] and [11] study the sum of monomials over rings); this is also something we are not considering here. For discussions on the size of term representation in supernilpotent algebras, see also [2].…”

Section: Introductionmentioning

confidence: 99%

The equation solvability problem over supernilpotent algebras with Mal’cev term

Kompatscher¹

2018

Int. J. Algebra Comput.

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In 2011 Horváth gave a new proof that the equation solvability problem over finite nilpotent groups and rings is in P. In the same paper he asked whether his proof can be lifted to nilpotent algebras in general. We show that this is in fact possible for supernilpotent algebras with a Mal'cev term. However, we also describe a class of nilpotent, but not supernilpotent algebras with Mal'cev term that have coNP-complete identity checking problems and NP-complete equation solvability problems. This proves that the answer to Horváth's question is negative in general (assuming P =NP).

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Section: Introductionmentioning

confidence: 99%

The equation solvability problem over supernilpotent algebras with Mal’cev term

Kompatscher¹

2018

Int. J. Algebra Comput.

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“…Here a polynomial is encoded by a string defining it, which in a finite algebra is proportional to its length (see e.g. [AMO17] for a precise definition). We remark that in the literature one can also find other ways of encoding polynomials, which might result in different complexities (e.g.…”

Section: Preliminariesmentioning

confidence: 99%

Notes on extended equation solvability and identity checking for groups

Kompatscher¹

2019

Acta Math. Hungar.

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Every finite non-nilpotent group can be extended by a term operation such that solving equations in the resulting algebra is NP-complete and checking identities is co-NPcomplete. This result was firstly proven by Horváth and Szabó; the term constructed in their proof depends on the underlying group. In this paper we provide a uniform term extension that induces hardness. In doing so we also characterize a big class of solvable, non-nilpotent groups for which extending by the commutator operation suffices.Date: August 24, 2018.

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“…If V is a variety with a Mal'tsev term, then k-supernilpotent algebras from V form a subvariety V s [3,Theorem 4.2]. An explicit set of identities axiomatizing V s relative to an axiomatization of V is known [3], but it is rather complicated, infinite, and unnatural in the context of groups and loops. Given a particular variety V, it is not clear from the general result of [3] whether a finite basis of V s exists.…”

Section: Introductionmentioning

confidence: 99%

“…An explicit set of identities axiomatizing V s relative to an axiomatization of V is known [3], but it is rather complicated, infinite, and unnatural in the context of groups and loops. Given a particular variety V, it is not clear from the general result of [3] whether a finite basis of V s exists. In groups, the question is of course settled by the equivalence of k-supernilpotence and k-nilpotence.…”

Section: Introductionmentioning

confidence: 99%